Elements of Algebra

CHAP. V. OF ALGEBRA. 13 CHAP. V. Of the Division of Simple Quantities. 45. When a number is to be separated into two, three, or more equal parts, it is done by means of division, which enables us to determine the magnitude of one of those parts. When we wish, for example, to separate the number $12$ into three equal parts, we find by division that each of those parts is equal to $4$. The following terms are made use of in this operation. The number which is to be decompounded, or divided, is called the dividend; the number of equal parts sought is called the divisor; the magnitude of one of those parts, determined by the division, is called the quotient: thus, in the above example, $12$ is the dividend, $3$ is the divisor, and $4$ is the quotient. 46. It follows from this, that if we divide a number by $2$, or into two equal parts, one of those parts, or the quotient, taken twice, makes exactly the number proposed; and, in the same manner, if we have a number to divide by $3$, the quotient taken thrice must give the same number again. In general, the multiplication of the quotient by the divisor must always reproduce the dividend. 47. It is for this reason that division is said to be a rule, which teaches us to find a number or quotient, which, being multiplied by the divisor, will exactly produce the dividend. For example, if $35$ is to be divided by $5$, we seek for a number, which multiplied by $5$, will produce $35$. Now, this number is $7$, since $5$ times $7$ is $35$. The manner of expression employed in this reasoning, is; $5$ in $35$ goes $7$ times; and $5$ times $7$ makes $35$. 48. The dividend therefore may be considered as a product, of which one of the factors is the divisor, and the other the quotient. Thus, supposing we have $63$ to divide by $7$, we endeavour to find such a product, that, taking $7$ for one of its factors, the other factor multiplied by this may exactly give $63$. Now $7 \times 9$ is such a product; and consequently $9$ is the quotient obtained when we divide $63$ by $7$. 49. In general, if we have to divide a number $ab$ by $a$, it is evident that the quotient will be $b$; for $a$ multiplied by $b$